Nearly all real-world measurements can only record a part of the underlying truth due to technical limitations. In many fields, full comprehension of the system requires an understanding of how the unmeasurable inputs or states map to the measurable outputs. In cases where many individual measurements are performed, the density of the observation can be approximated with histograms. They count the frequency at which measurements fall in a given range. Each observed sample corresponds to exactly one unknown point in the input space that has been mapped by a function to produce exactly this recorded output. When the distribution of these points in the original input space is known (e.g. uniformly distributed), a transport function describing this mapping can be found. Identifying this transport function is the main objective of this thesis. The field of transportation theory is dedicated to finding these transportation maps between two (probability) measures that are optimal according to a metric. Those approaches can fail to identify the true underlying transport map, for example if it is not bijective or when the recorded density is discontinuous. Reconstructing this true underlying transport map can be done by employing an observation process that measures consecutive outputs of moving points. This reconstruction procedure is implemented with artificial neural networks and demonstrated by examples. Separately to the transport of measures, another network is implemented that learns the underlying dynamical system based on the observation process, allowing to extrapolate the movement of the points. Apart from fictitious examples, the procedure is also applied to reconstruct the shape of a simulated cell by synthesizing image data (e.g. produced by a microscope) and observing moving bacteria on the cell's surface.     «

Nearly all real-world measurements can only record a part of the underlying truth due to technical limitations. In many fields, full comprehension of the system requires an understanding of how the unmeasurable inputs or states map to the measurable outputs. In cases where many individual measurements are performed, the density of the observation can be approximated with histograms. They count the frequency at which measurements fall in a given range. Each observed sample corresponds to exactly...     »